Describing Motion

Scalar & Vector Quantities

• All quantities can be one of two types:
  • A scalar
  • A vector

Scalars

• Scalars are quantities that have only a magnitude
  • For example, mass is a scalar since it is a quantity that has magnitude without a direction
  • Distance is also a scalar since it only contains a magnitude, not a direction

Vectors

• Vectors have both magnitude and direction
• Velocity, for instance, is a vector since it is described with both a magnitude and a direction
  • When describing the velocity of a car it is necessary to mention both its speed and the direction in which it is travelling
  • For example, the velocity might be 60 km per hour (magnitude) due west (direction)
• Distance is a value describing only how long an object is or how far it is between two points – this means it is a scalar quantity
• Displacement on the other hand also describes the direction in which the distance is measured – this means it is a vector quantity
  • For example, a displacement might be 100 km north

Examples of Scalars & Vectors

• The table below lists some common examples of scalar and vector quantities:

Scalars & Vectors Table

• Some vectors and scalars are similar to each other
  • For example, the scalar quantity distance corresponds to the vector quantity displacement
• Corresponding vectors and their scalar counterparts are aligned in the table where applicable

Comparing Scalars & Vectors

• The worked example below illustrates how to determine whether a quantity is a scalar or a vector

Step 1: Recall the definitions of a scalar and vector quantity

• • Scalars are quantities that have only a magnitude
  • Vectors are quantities that have both magnitude and direction

Step 2: Identify which quantity has magnitude only

• • Mass is a quantity with magnitude only
  • So mass is a scalar quantity
    • Blu might explain to his junior astronauts that their mass will not change if they travel to outer space

Step 3: Identify which quantity has magnitude and direction

• Weight is a quantity with magnitude and direction (it is a force)
• So weight is a vector quantity
  • Blu might explain that to his junior astronauts that their weight – the force on them due to gravity – will vary depending on their distance from the centre of the Earth

Calculating Speed

• The speed of an object is the distance it travels every second
• Speed is a scalar quantity
  • This is because it only contains a magnitude (without a direction)
• For objects that are moving with a constant speed, use the equation below to calculate the speed:

• Where:
  • Speed is measured in metres per second (m/s)
  • Distance travelled is measured in metres (m)
  • Time taken is measured in seconds (s)

Calculating Average Speed

• In some cases, the speed of a moving object is not constant
  • For example, the object might be moving faster or slower at certain moments in time (accelerating and decelerating)
• The equation for calculating the average speed of an object is:

How to Use Formula Triangles

• Formula triangles are really useful for knowing how to rearrange physics equations
• To use them:

1. Cover up the quantity to be calculated, this is known as the ‘subject’ of the equation
2. Look at the position of the other two quantities
   • If they are on the same line, this means they are multiplied
   • If one quantity is above the other, this means they are divided – make sure to keep the order of which is on the top and bottom of the fraction!

• In the example below, to calculate speed, cover-up ‘speed’ and only distance and time are left
  • This means it is equal to distance (on the top) ÷ time (on the bottom)

Step 1: List the known quantities

• • Average speed = 250 m/s
  • Time taken = 2 hours

Step 2: Write the relevant equation

Step 3: Rearrange for the total distance

total distance = average speed × time taken

Step 4: Convert any units

• • The time given in the question is not in standard units
  • Convert 2 hours into seconds:

2 hours = 2 × 60 × 60 = 7200 s

Step 5: Substitute the values for average speed and time taken

total distance = 250 × 7200 = 1 800 000 m

Velocity

• The velocity of a moving object is similar to its speed, except it also describes the object’s direction
  • The speed of an object only contains a magnitude – it’s a scalar quantity
  • The velocity of an object contains both magnitude and direction, e.g. ‘15 m/s south’ or ‘250 mph on a bearing of 030°’
• Velocity is therefore a vector quantity because it describes both magnitude and direction

Distance-Time Graphs

• A distance-time graph shows how the distance of an object moving in a straight line (from a starting position) varies over time

Constant Speed on a Distance-Time Graph

• Distance-time graphs also show the following information:
  • If the object is moving at a constant speed
  • How large or small the speed is

• A straight line represents constant speed
• The slope of the straight line represents the magnitude of the speed:
  • A very steep slope means the object is moving at a large speed
  • A shallow slope means the object is moving at a small speed
  • A flat, horizontal line means the object is stationary (not moving)

Changing Speed on a Distance-Time Graph

• Objects sometimes move at a changing speed
  • This is represented by a curve
• In this case, the slope of the line will be changing
  • If the slope is increasing, the speed is increasing (accelerating)
  • If the slope is decreasing, the speed is decreasing (decelerating)
• The image below shows two different objects moving with changing speeds

Gradient of a Distance-Time Graph

• The speed of a moving object can be calculated from the gradient of the line on a distance-time graph:

• The rise is the change in y (distance) values
• The run is the change in x (time) values

Step 1: Draw a large gradient triangle on the graph and label the magnitude of the rise and run

• • The image below shows a large gradient triangle drawn with dashed lines
  • The rise and run magnitude is labelled, using the units as stated on each axis

Step 2: Convert units for distance and time into standard units

• • The distance travelled (rise) = 8 km = 8000 m
  • The time taken (run) = 6 mins = 360 s

Step 3: State that speed is equal to the gradient of a distance-time graph

• • The gradient of a distance-time graph is equal to the speed of a moving object:

Step 4: Substitute values to calculate the speed

speed = gradient = 8000 ÷ 360

speed = 22.2 m/s